In mathematics, a **limit** is a fundamental concept in calculus and analysis that describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential for defining derivatives, integrals, and continuity.

### Basic Definition of a Limit

For a function \( f(x) \), the limit as \( x \) approaches a value \( a \) is written as:

\[
\lim_{{x \to a}} f(x) = L
\]

This means that as \( x \) gets arbitrarily close to \( a \) (from either side), the value of \( f(x) \) approaches \( L \). Here, \( L \) is the limit of \( f(x) \) as \( x \) approaches \( a \).

### One-Sided Limits

- **Left-hand limit:** The limit of \( f(x) \) as \( x \) approaches \( a \) from the left (i.e., values less than \( a \)) is denoted by:

\[
\lim_{{x \to a^-}} f(x)
\]

- **Right-hand limit:** The limit of \( f(x) \) as \( x \) approaches \( a \) from the right (i.e., values greater than \( a \)) is denoted by:

\[
\lim_{{x \to a^+}} f(x)
\]

A limit exists if and only if both the left-hand and right-hand limits exist and are equal.

### Examples

1. **Finite Limit:**  
   Consider \( \lim_{{x \to 2}} (3x + 1) \).  
   As \( x \) approaches 2, the expression \( 3x + 1 \) approaches \( 3 \cdot 2 + 1 = 7 \). Thus,  
   \[
   \lim_{{x \to 2}} (3x + 1) = 7.
   \]

2. **Infinite Limit:**  
   Consider \( \lim_{{x \to 0}} \frac{1}{x^2} \).  
   As \( x \) approaches 0, \( \frac{1}{x^2} \) grows without bound. Thus,  
   \[
   \lim_{{x \to 0}} \frac{1}{x^2} = \infty.
   \]

3. **Limit Does Not Exist (DNE):**  
   Consider \( \lim_{{x \to 0}} \frac{1}{x} \).  
   As \( x \) approaches 0 from the right, \( \frac{1}{x} \) tends to \( +\infty \), but as \( x \) approaches 0 from the left, it tends to \( -\infty \). Since the left-hand limit and the right-hand limit are not equal, the limit does not exist.

### Tips for Calculating Limits

1. **Direct Substitution:** Try plugging in the value of \( x \) directly. If the function is continuous at that point, this gives the limit.
2. **Factorization:** If direct substitution results in an indeterminate form like \( \frac{0}{0} \), try factoring the numerator and denominator to cancel out terms.
3. **Rationalization:** For limits involving square roots, multiply by a conjugate to simplify.
4. **L'Hopital's Rule:** If you encounter indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), differentiate the numerator and the denominator separately.
5. **Squeeze Theorem:** If a function is "squeezed" between two other functions that have the same limit at a point, then it has the same limit at that point.

### Would you like more details on a specific type of limit or a related concept?

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### Related Questions:

1. What are some common indeterminate forms in limits?
2. How do you compute the limit of a sequence?
3. What is the formal epsilon-delta definition of a limit?
4. How does continuity relate to limits?
5. How do you find limits involving trigonometric functions?

**Tip:** To master limits, practice solving a variety of problems using different methods, such as factorization, L'Hopital's rule, and the squeeze theorem.

Explore the fundamental concept of limits in calculus, including finite and infinite limits, and cases where limits do not exist. Learn how to apply key methods such as L'Hopital's Rule and the Squeeze Theorem. Suitable for high school students in Grades 11-12 studying pre-calculus or calculus.

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